Vector groups in Ogus-Vologodskys Nonabelian Hodge theory in characteristic p

Question

i hope this is the right place for this. In https://math.berkeley.edu/~ogus/preprints/anonhodge.pdf on page 15 Ogus and Vologodsky state the following:

Let $\pi_T: \textbf{T} \rightarrow X$ be a vector group and let $T$ be its sheaf of sections. Thus $T$ is a locally free sheaf of $\mathcal{O}_X$-modules with dual $\Omega$ and $\textbf{T}=Spec_X(Sym(\Omega))$. The pairing $T \times \Omega \rightarrow \mathcal{O}_X$ extends to a pairing $T \times Sym(\Omega) \rightarrow Sym(\Omega)$, where sections of $T$ act as derivations on $Sym(\Omega)$. The action defines a map $\xi \mapsto D_{\xi}: T \rightarrow {\pi_T}_{\star} T_{\textbf{T}/X}$ which identifies T with the sheaf of translation invariant vector fields of $\textbf{T}$ relative to $X$. It also induces an isomorphism $ \pi_T ^{\star}T \rightarrow T_{\textbf{T}/X}$.

Question: First of all the phrase "vector group" (in this context) is unfamiliar to me, for me it looks like a vector bundle. Then i do not understand how we extend $T \times \Omega \rightarrow \mathcal{O}_X $ to $T \times Sym(\Omega) \rightarrow Sym(\Omega)$ such that

1. sections of $T$ act as derivations on $Sym(\Omega)$, and

2. $\xi \mapsto D_{\xi}$ (what is $D_{\xi}$ here?) identifies $T$ with the sheaf of translation invariant vector fields of $\textbf{T}$.

My guess at this point is that I am misinterpreting the term "vector group", which I know to be a term in Lie theory. This, and the fact that we know that the space of translation invariant vector fields of a Lie group can be identified with the tangent space (at the identity), make me believe the answer to lie in this direction. Any help would be very much appreciated.

Answer 1

Arthur Ogus was so kind as to explain this to me in an Email:

The use of the phrase "vector group" was motivated by the similarities to Lie theory (as stated in my question).

As to the construction: For every $\xi \in T$ we get a map $ Sym^k( \Omega) \rightarrow Sym^{k-1} ( \Omega)$ by interior multiplication by $\xi$, adding these up we get a map $D_{\xi}: Sym( \Omega) \rightarrow Sym(\Omega)$ which is a derivation.